On the Analogy of Processes in Thermodynamic and Microeconomic Systems

Abstract

This work states the typical problems in thermodynamic optimization. The authors present an overview of the results of studies focused on the ultimate capabilities of macrosystems in thermodynamics and microeconomics, taking into account the irreversibility of the processes occurring in them. The research methodology is based on adding an entropy balance to energy and matter balances. This allows for the refining of reversible indicators, such as the reversible efficiency coefficient, by accounting for kinetic factors, such as transfer coefficients, which indirectly reflect the size of devices, kinetic equation forms, and others. For processes that use heat energy, the set of feasible solutions within the ‘target flow intensity–energy expenses’ plane is convex upwards and limited. This paper also provides conditions for the minimum dissipation of processes at a given intensity. These conditions define the boundary of the feasibility set. Finally, this paper compares and lists the similarities between thermodynamic and microeconomic systems and demonstrates the ultimate capabilities of an intermediary in microeconomic systems and the optimal parameters of a working medium in thermodynamic systems. These are divided into active and passive subsystems. The latter, in turn, can have finite and infinite capacity (reservoirs).

1. Introduction

When Sadi Carnot solved the problem of the maximum efficiency factor of heat-to-work converters, he kickstarted thermodynamics. One of his contemporaries soon posed the problem of the maximum power of such a converter. But it took more than 200 years before the power limit problem became relevant. In the nuclear power industry, the cost of fuel is considerably lower than in the coal and gas industries, but the cost of equipment is significantly higher. This is why it is important to obtain maximum power from a thermal machine in nuclear power engineering, which means that neither fuel nor efficiency are of the essence. In 1957, Novikov’s work appeared in the Atomic Energy journal [1]. It was devoted to the heat machine cycle corresponding to the maximum of its power. Their work assumed a priori that such a cycle consists of two isotherms and two adiabats; their work determined the temperatures of a working body in contact with hot and cold sources and the efficiency corresponding to these values. Naturally, Novikov’s efficiency turned out to be significantly lower than that of Carnot. Novikov’s work was not known to many. Only in 1975 did Curzon and Alburn [2] independently solve the same problem. Their publication gave rise to a stream of research devoted to the evaluation of the limits of the most different thermodynamic systems (thermal, refrigeration, and diffusion machines, gas and liquid separation systems, heat and mass transfer systems, chemical systems). Among other significant contributions are the works of R.S. Berry, P. Salamon, B. Andresen [3,4,5,6,7], and K.H. Hoffmann [8]. The limits were understood as either the coefficients of conversion of the energy input into the target flow, or the limits of this target flow. Naturally, the flows were non-zero; thus, the processes were irreversible. The kinetic function was quite important—it showed how the flows depended on the driving forces, in particular, heat and mass transfer coefficients related to the size of the apparatus.

Today, these studies have formed a new direction in thermodynamics called finite-time thermodynamics, studied by P. Salamon. This name stems from the first papers that studied non-stationary processes in which a given amount of product needed to be obtained for a limited time. Subsequently, the majority of studies were related to non-equilibrium stationary processes, in which time is of no importance at all. Therefore, the authors believe that “optimization thermodynamics” proposed by L. I. Rosonoer is more in line with the challenges it addresses.

Thermodynamic systems represent only one of the most studied varieties of macrosystems. Their research methodology can be applied to macrosystems of other species, considering their specific aspects. In this paper, the results and methods of optimization thermodynamics are applied to thermodynamic and microeconomic systems. This paper demonstrates the similarities in statements and solutions to problems of both systems.

2. Mathematical Models of Macrosystems

Macrosystems (MS) are systems consisting of a large number of elements, each of which cannot be controlled. Such systems are managed only at the macro level. Examples include thermodynamic systems, microeconomics, population migration, segregated systems, etc.

In the case of segregated systems, there are a large number of cells (aggregates) [9] interacting with each other through a common medium. It is only possible to control the parameters of the environment.

2.1. Models of Macrosystems, Quantitative Measure of Irreversibility, Typical Problems and Methods for Their Solution

In the phenomenological approach to macrosystems, their state is characterized by two types of variables: extensive and intensive. The former are doubled when two identical systems are combined, while the latter remain unchanged. Examples of extensive variables include volume, internal energy, and matter quantity in thermodynamics, and resource and capital stocks in economics. Examples of intense variables include temperature, pressure, and chemical potentials in thermodynamics, and resource prices in economics. A macrosystem is in equilibrium if, for any part of it that includes sufficient trace elements, the intensive variables are the same. For such a system, the variables are related to each other by the equation of state. For example, the ideal gas law is

$$pV=NRT,$$

where $p$ and $V$ are the gas pressure and volume, $N$ is the number of moles, $T$ is the absolute temperature, and $R$ is the universal gas constant that reflects molar heat capacity.

Equilibrium macrosystems can be divided into three types: infinite-capacity systems, in which intensive variables are constant or vary over time irrespective of the values of extensive variables (thermodynamic reservoirs, perfect competition markets); finite-capacity systems, in which the intensive variables depend on the extensive variables (thermodynamic systems of finite volume, economic systems with finite resource reserves); and active systems, in which the values of intensive variables can be controlled independently of extensive ones (working body of a heat machine, middleman or production company, dealer in financial markets).

It is the intensive variables of active systems that are the driving influences in the optimization problems of macrosystems.

A macrosystem is considered closed if it is isolated from the environment; otherwise, it is considered open. The system can be isolated for only some of the variables, not necessarily for all of them.

An important feature of macrosystems is that when they interact, there are exchange processes that lead to the flows changing the extensive variables of the systems, so that in finite-capacity systems, the intensive variables are equalized. These natural processes take place without any environmental influence. To return the interacting systems to its initial state, some kind of resource (energy, usually) is required from the environment. Such processes are called irreversible.

One of the distinct features of macrosystems is the quantitative measure of the irreversibility of processes, which, when changed, reflects the amount of resource needed to bring macrosystems back to their initial state after an irreversible process. In thermodynamics, the measure of irreversibility is entropy; in economics, it is the welfare function, which will be elaborated on below. Both are denoted further by $S$. For any process in a macrosystem, the measure of irreversibility does not decrease, and if it increases, its rate of increase is called dissipation.

In the study of irreversible processes in macrosystems, they are divided into separate equilibrium subsystems, so that the change in the irreversibility index is associated with the exchange processes between these subsystems. Such an assumption (the replacement of a system with distributed parameters by a set of systems with concentrated parameters) allows the researchers to use equations of state and variables that make sense only in equilibrium. This approach is used in thermodynamic optimization [5,6,7,8,9]:

The system is split into equal subsystems. Each subsystem is given its own by-substance and by-energy balance equations. Streams appear at the borders of the subsystem interaction, and the irreversibility increases [10,11,12,13,14,15]. In thermodynamics, this means that the entropy in the whole system increases; in economics, the welfare function of each of the contacting subsystems increases instead [16,17,18,19,20,21].

Let us describe typical optimization problems of irreversible thermodynamic and microeconomic systems and the general methodology of their solution, and then, give their quantitative formulations and solution results. The problems are as follows:

• Introduce a quantitative measure of irreversibility for economic macrosystems, taking into account their specific aspects.
• Establish conditions with fitting processes that are marginally irreversible under certain constraints—minimum dissipation processes.
• Find the distribution of values of extensive variables in the steady-state mode of an open macrosystem, including an active subsystem.
• Find the optimal strategy of an active system in a closed and in an open macrosystem.
• Define the area of feasible states of a macrosystem.

The last three classes of problems are closely related to the first two, since it turns out that a macrosystem reaches its limits when the processes occurring in it are the processes of minimum dissipation.

Some of these problems are solved for all or specific types of thermodynamic systems; thus, this work is mainly aimed at stating and solving these problems in microeconomics.

2.2. Quantitative Measure of Irreversibility for Microeconomic Systems

Let $N\in R^{n+1}$ be the vector of resources of an economic system (economic agent (EA)), one component of which $N_0$ is the basic resource—capital. Let $p_i$ be the valuation of the i-th resource—the minimum price at which the EA is willing to sell and the maximum price at which it is willing to buy the i-th resource. When selling (buying) the resource at price $p_i$, its flow is as small as possible, so $p_i$ is also called the equilibrium price.

The following statement holds: there exists a function $S(N)$ such that its differential is as follows:

$$dS=p_0(N)\left[d{N}_0+{\displaystyle \sum _{i=1}^n{p}_i(N)d{N}_i}\right].$$

(1)

This function is called the welfare function. Its existence for $n=1$ is proven by Rosonoer in [22,23,24] and for $n>1$ in [9]. The existence of the welfare function is very important, because the assumption that it is twice continuously differentiable leads to the equations that define prices in equilibrium for EA resources and capital, as well as the link between the sensitivity of these prices and the change in the resource reserves:

$$p_0=\frac{\mathsf{\partial }S}{\mathsf{\partial }{N}_0}>0,$$

(2)

$${\overline{p}}_i=p_0\cdot p_i=\frac{\mathsf{\partial }S}{\mathsf{\partial }{N}_i},$$

(3)

$$p_i=\frac{\mathsf{\partial }S/\mathsf{\partial }{N}_i}{\mathsf{\partial }S/\mathsf{\partial }{N}_0},$$

(4)

$$\frac{\mathsf{\partial }{\overline{p}}_i}{\mathsf{\partial }{N}_j}=\frac{\mathsf{\partial }{\overline{p}}_j}{\mathsf{\partial }{N}_i}=\frac{{\mathsf{\partial }}^{2}S}{\mathsf{\partial }{N}_i\mathsf{\partial }{N}_j}.$$

(5)

For all exchange processes in the microeconomy, the total welfare function does not decrease. Moreover, in thermodynamics, only the total entropy of the system does not decrease, but in economics, the welfare function of each EA does not decrease either. The last statement is called the voluntary exchange principle.

When combining two equal microeconomic systems, the welfare function is doubled along with each of its arguments. This means that this homogeneous first-order function can be transformed, as per Euler’s theorem, to

$$S(N)=p_0\left(N_0+{\displaystyle \sum _i{p}_i{N}_i}\right).$$

(6)

The total welfare function never decreases in any exchange processes in microeconomics. Unlike in thermodynamics, where only the total entropy of a system never decreases, in economics, the welfare function of each EA does not decrease either. This statement is called the principle of voluntariness.

It follows from the principle of voluntariness that two EAs can exchange one i-th type of resource only when its valuations $p_{1i}$ and $p_{2i}$ have different signs, i.e., for one EA, the resource is a burden, and for another, it is useful. Otherwise, in order for inequalities $d{S}_1\ge 0$ and $d{S}_2\ge 0$ to be simultaneously fulfilled for non-zero valuations, there must be an exchange of at least two flows. The exchange flow between EAs $g$ is zero in equilibrium when the valuation vectors are equal to each other $(p_1=p_2)$. If the active subsystem exchanges with the EA such that the state of the EA in space $N$ changes cyclically, then

$${\displaystyle \oint dS}\ge 0.$$

(7)

The equality in (7) corresponds to an exchange at equilibrium prices, and its duration is $\tau \to \infty$. The economic reservoir welfare function is as follows:

$$S_m=p_0\left(N_0+{\displaystyle {\sum }_{i=1}^n{p}_i\cdot N_i}\right),$$

(8)

where $p_0$ and $p_1$ are constants or functions of time (non-stationary market). For an active subsystem (intermediary),

$$S_a=p_0\left(N_0\right)\cdot N_0.$$

(9)

The purpose of exchange is to extract capital.

Let the intermediary make an exchange with the EA for a finite time $\tau$ in such a way that the state of the EA makes a cycle. The resource flow $N$ (scalar, for simplicity) has the same sign as the difference between the price $C$ and the resource valuation $p$

$$Sign g(C,p)=Sign (C-p).$$

(10)

EA’s capital reserves at the final step of the process and the increase in the welfare function are calculated as follows:

$$\overline{N}=N_0+p\left(N\right)\cdot N,$$

(11)

$$\Delta S=p_0{\displaystyle \underset{0}{\overset{\tau }{\int }}g(C,p)\cdot (C-p)dt>0}.$$

(12)

For the intermediary, the expenses of capital are

$$\Delta N_{0a}=-{\displaystyle \underset{0}{\overset{\tau }{\int }}g(C,p)\cdot Cdt}.$$

(13)

If the intermediary had purchased the same amount of resource at reversible prices, its costs would have been less.

$$\Delta N_{0a}^0=-{\displaystyle \underset{0}{\overset{\tau }{\int }}g(C,p)\cdot pdt}.$$

(14)

The capital loss (trade costs) is

$$\Delta N_{0a}^0-\Delta N_{0a}^{}={\displaystyle \underset{0}{\overset{\tau }{\int }}g(C,p)\cdot (C-p)dt}=\Delta S.$$

(15)

Thus, the increase in the welfare function is the trade costs, economically speaking. The increasing loss at the current moment in time is represented by

$$\sigma =g(C,p)\cdot (C-p)$$

(16)

and is called capital dissipation.

3. Minimal Dissipation Processes

3.1. Thermodynamics

Let us consider the exchange process of the finite-capacity subsystem and the active subsystem of duration $\tau$. Flow $g\in R^1$ depends only on the intensive variables $U_1(N)$ and $U_2(t)$. $U_1$ depends on the extensive variable $N_1$, and $U_2$ is the control. The entropy production is the product of the flow $g(U_1,U_2)$ and the driving force $x(U_1,U_2)$. The change rate of the extensive variable $N$ depends on the flow, and due to the state equation, the variable $U_1$ also changes, with its rate of change depending on $U_1$ and $U_2$. The average intensity of the flow is given. The minimal dissipation problem is

$$\overline{\sigma }=\frac{1}{\tau }{\displaystyle \underset{0}{\overset{\tau }{\int }}g(U_1,U_2)\cdot x(U_1,U_2)dt}\to \underset{U_2}{\min }$$

(17)

when

$$\frac{d{U}_1}{dt}=\phi (U_1,U_2),$$

(18)

$$U_1(0)=U_{10},$$

(19)

$$\mathsf{\phi }\ne 0,$$

(20)

$$\frac{1}{\tau }{\displaystyle \underset{0}{\overset{\tau }{\int }}g(U_1,U_2)dt}=\overline{g}.$$

(21)

The optimal condition of this problem is obtained in [9] for one particular case when $\phi (U_1,U_2)=\alpha (U_1)\cdot g(U_1,U_2)$. In this case,

$$\frac{g^2}{\mathsf{\partial }g/\mathsf{\partial }{U}_2}\cdot \frac{\mathsf{\partial }x}{\mathsf{\partial }{U}_2}=const.$$

(22)

For heat transfer, when the heat flow $q$ is Newtonian,

$$\begin{gathered} T_1~U_1, T_2~U_2, g~q=\alpha \cdot (T_1-T_2), \\ \phi (T_1,T_2)=\frac{q}{C(T_1)}, x=\left(\frac{1}{T_2}-\frac{1}{T_1}\right), \end{gathered}$$

where $C$ is heat capacity. It follows from condition (22) that at any time, the temperature ratio $T_1/T_2$ must be constant. For the radiant heat transfer, when the heat flow is proportional to the difference in the fourth powers of temperatures, at each moment in time, the following expression should be constant (see [9]):

$$\frac{T_2^4-T_1^4}{T_2^{5/2}}=const.$$

(23)

If a two-dimensional stream of heat and substance appears between subsystems, the conditions of minimum dissipation require two equations to be constant instead of one. Both equations depend on intensive variables of interacting subsystems.

The solution of problem (19)–(21) becomes substantially simpler when the flows are linearly related to the driving forces (Onsager relations) and lead to the fact that the vector of driving forces (and hence, the vector of flows) must be constant at any time.

3.2. Microeconomics

Since capital dissipation is the intensity of the intermediary’s losses when buying a resource (overpayment) or selling it (discount) relative to equilibrium prices, the minimum dissipation corresponds to a minimum of EA capital when finishing buying (selling) the resource, if the quantity $\Delta N$ of the resource and the duration of the process are given. For the scalar resource, the problem is

$$N_0(\tau )\to \underset{C(t)}{\min }$$

(24)

when

$$\frac{d{N}_0}{dt}=C\cdot g(p,C),$$

(25)

$$N_0(0)=N_0^0,$$

(26)

$$\frac{dN}{dt}=-g(p,C),$$

(27)

$$N(0)=N_0,$$

(28)

$${\displaystyle \underset{0}{\overset{\tau }{\int }}g(p,C)dt}=\Delta N.$$

(29)

Here, $C(t)$—intermediary price and $p=p(N_0,N)$—equilibrium price of EA.

Let us demonstrate the result of the solution for this problem. The minimum dissipation condition is as follows:

$$\frac{d}{dN}\left[\frac{\mathsf{\partial }g/\mathsf{\partial }C}{g^2}\right]=\frac{\left(\mathsf{\partial }g/\mathsf{\partial }p\right)\cdot \left(\mathsf{\partial }p/\mathsf{\partial }{N}_0\right)}{g^2}.$$

(30)

This condition coupled with condition (29) determine $C^{\ast }\left(N_0,N\right)$. If the price in equilibrium does not depend on the supply of capital, i.e., $\mathsf{\partial }p/\mathsf{\partial }{N}_0=0$, this condition leads to the following equation:

$$\frac{\mathsf{\partial }g/\mathsf{\partial }C}{g^2}=const.$$

(31)

In particular, if the supply function is linear $g=\alpha \cdot \left(C-p(N)\right)$, it follows from (31) that $g(C^{\ast },p)=const \forall t$.

Example 1.

A company needs to buy a quantity of resource M from an EA in a constant time τ. The supply function and dependency of the EA’s price in equilibrium from resource reserves are as follows:

$$g=\alpha \cdot \left(C-p(N)\right), p(N)=\frac{k}{\sqrt{N}}.$$

From condition (31), $C=p(N)+\lambda$, where λ is a constant.

The integral of the supply function from 0 to τ is M; therefore, the constant $\lambda =\frac{M}{\alpha \tau }$. The resource reserves of EA change in time as

$$N(t)=N(0)-\frac{k}{N(0)-\frac{M}{\alpha \tau }t}.$$

The optimal price is

$$C^{\ast }(t)=\frac{M}{\alpha \tau }+\frac{k}{\sqrt{N(0)-\frac{M}{\alpha \tau }t}}.$$

This increases while the resource is being bought.

The supply function and the dependence of the market’s equilibrium price from the resource reserves may be unknown to a buyer, but they can be reconstructed from experimental data on the market’s behavior when purchasing prices change.

4. Stationary State of Open Systems and Intermediary’s Marginal Capacity

4.1. Thermodynamics

Let us consider an open system (Figure 1) consisting of m subsystems with internal equilibrium, two reservoirs, and an active subsystem. There are flows between all subsystems, depending on the difference in their intensive variables, so that the system as a whole is not in equilibrium.

In stationary mode, the flows are non-zero if the number of reservoirs is greater than two, and their intensive variables $p_{+}$, $p_{-}$ are different. For simplicity, let us consider a system with two reservoirs. The active system can interact with either the reservoirs or any of the subsystems by setting the intensive variables $U_{+}$, $U_{-}$, $U_i$$(i=1,\dots ,m-2)$ at the contact points. Let us consider temperature as the first component of each of vectors $p$ and $U$, so $T_i=p_{1i}$, $T_{ai}=U_{1i}$.

Finally, let us choose values of variables $U$ and vectors $p_i$$(i=1,\dots ,m-2)$ such that the “organized” energy flow $N$ (mechanical work, separation energy, electrical energy) extracted by the system is maximal. Let $q$ be energy flows and $g$ be material flows.

The problem, formally, is

$$N={\displaystyle \sum _{i=1}^m{q}_i(p_i,U_i)}\to \underset{U,p}{\max }$$

(32)

when the balance relations of energy, substance, and entropy for the (m − 2)-th subsystems and converter are fulfilled

$${\displaystyle \sum _{j=1}^m{q}_{ij}(p_i,p_j)}=q_i(p_i,U_i), i=1,\dots ,m-2,$$

(33)

$${\displaystyle \sum _{j=1}^m{g}_{ij}(p_i,p_j)}=g_i(p_i,U_i), i=1,\dots ,m-2,$$

(34)

$${\displaystyle \sum _{j=1}^n\left[q_{ij}(p_i,p_j)\cdot s_{ij}+\frac{q_{ij}(p_i,p_j)}{p_{1i}}\right]}=0,i=1,\dots ,m-2,$$

(35)

$${\displaystyle \sum _{i=1}^m{g}_i(p_i,U_i)}=0,$$

(36)

$${\displaystyle \sum _{i=1}^m\left[g_i(p_i,U_i)\cdot s_i+\frac{q_i(p_i,U_i)}{U_{1i}}\right]}=0.$$

(37)

Here, $s_{ij}$ and $s_i$ are the molar entropies of flows $g_{ij}$ and $g_i$, respectively; generally speaking, they may depend on $U$ and $p$.

The formulated problem is considered a nonlinear programming problem. Its solution determines the limiting power of the converter and the distribution of matter and energy in the open system. The chemical reactors in it will add appropriate sum items to the material and entropic balance conditions [9]. For a particular case of the problem (32)–(37), when only heat exchange took place, and the heat flows were Newtonian with coefficients ${\alpha }_{ij}$ and ${\alpha }_i$, respectively, in [9], optimal conditions were obtained in the form of

$${\displaystyle \sum _{i=1}^m{\alpha }_i\cdot \frac{T_i}{U_i}}={\displaystyle \sum _{i=1}^m{\alpha }_i}, i=1,\dots ,m,$$

(38)

$$U_i^2=\Lambda \cdot \frac{T_i}{1-{\lambda }_i}, i=1,\dots ,m,$$

(39)

$${\alpha }_i\left(1+\frac{\Lambda }{U_i}-{\lambda }_i\right)={\lambda }_i{\displaystyle \sum _{i=1}^m{\alpha }_{ij}}, i=1,\dots ,m.$$

(40)

In particular, when $m=2$, $T_1=T_{+}$, $T_2=T_{-}$, $U_1^{\ast }=k\sqrt{T_{+}},$$U_2^{\ast }=k\sqrt{T_{-}}$, the machine efficiency in this case is

$$\eta =1-\sqrt{\frac{T_{-}}{T_{+}}},$$

(41)

$$N_{\max }=\frac{{\alpha }_1\cdot {\alpha }_2}{{\alpha }_1+{\alpha }_2}{\left(\sqrt{T_{+}}-\sqrt{T_{+}}\right)}^2.$$

(42)

For the vector flows $g$, linearly dependent on the driving forces $x$, the Onsager reciprocity conditions are valid, and for any vector $U$ of intensive variables of the active subsystem, the values of the extensive variables are distributed in such a way that the total entropy production is minimal.

$$\sigma =\frac{1}{2}{\displaystyle \sum _{i,j=1}^m{x}_{ij}^T\cdot A_{ij}\cdot x_{ij}}+{\displaystyle \sum _{i=1}^m{x}_i^T\cdot A_i\cdot x_i}\to \underset{p}{\min }.$$

(43)

Here, $x_i(p_i,U_i)$ is the vector of driving forces from i-th subsystem to the converter; $x_{ij}(p_i,p_j)$ is the same, but from i-th subsystem to the j-th. $A_{ij}$ are the elements of the matrix of kinetic coefficients, where $A_{ij}=A_{ji}$.

The stated rule is the extreme Prigozhin principle for a system with a converter.

4.2. Microeconomics

For resource flows linearly dependent on the difference between valuations and prices, and for an EA with a welfare function $S(N)$, the reciprocity conditions are valid [25,26,27,28].

Let the flow of the i-th resource be

$$g_i={\displaystyle \sum _{j=1}^m{\alpha }_{ij}\cdot {\Delta }_j=A\cdot {\Delta }_i}, i=1,\dots ,k,$$

(44)

$${\Delta }_j=C_j-p_j, i=1,\dots ,k.$$

(45)

Capital dissipation is equal to

$$\sigma ={\displaystyle \sum _{i=1}^k{g}_i\cdot {\Delta }_i}=\frac{d}{dt}\left[\frac{S}{p_0}\right]=g^T\cdot B\cdot g,$$

(46)

$$B=A^{-1}.$$

(47)

The elements $b_{ij}$ of matrix $B$ are determined by the second derivative $\frac{\mathsf{\partial }}{\mathsf{\partial }{N}_i\mathsf{\partial }{N}_j}\left[\frac{S}{p_0}\right]$, which does not depend on the order of differentiation, i.e., $b_{ij}=b_{ji}$. This means that the following matrix is also symmetric to $A$.

$$a_{ij}=a_{ji}.\forall \hspace{0.17em}\hspace{0.17em}i,j.$$

(48)

The structure of an open microeconomic system with an intermediary coincides with that of Figure 1. Let $U_i$ be the price of purchase (sale) of the resource by the intermediary to the i-th EA; let n be the intensity of the capital flow extracted from the system; let $U_i$ be the flow of resource between the intermediary and EA, depending on its evaluation $p_i$ and purchase price; let $g_{ij}(p_i,p_j)$ be the flow of resource exchange between EAs. The problem, formally, is

$$n=-{\displaystyle \sum _{i=1}^m{g}_i(p_i,U_i)\cdot U_i}\to \underset{U,p}{\max }$$

(49)

when balancing resources for each EA except for reservoirs

$${\displaystyle \sum _{j=1}^m{g}_{ij}(p_i,p_j)}=g_i(p_i,U_i), i=1,\dots ,m-2,$$

(50)

and as long as the intermediary does not accumulate resources

$${\displaystyle \sum _{i=1}^k{g}_i(p_i,U_i)}=0.$$

(51)

The minus sign in (49) is explained by the fact that the positive direction of flow $g$ is taken to be the direction from the EA to the intermediary, which is accompanied by capital costs.

The problem (49)–(51) is a standard nonlinear programming problem, the optimal conditions of which come from the Kuhn–Tucker theorem. For $g_{ij}={\alpha }_{ij}\cdot (p_j-p_i)$, $g_i={\alpha }_i\cdot (U_i-p_i)$; these are:

$$U_i=0.5\cdot (p_i+{\lambda }_i+\Lambda ), i=1,\dots ,m,$$

(52)

$$\Lambda -U_i+{\lambda }_i=-\frac{{\lambda }_i}{{\alpha }_i}{\displaystyle \sum _{j=1}^m{\alpha }_{ji}}, i=1,\dots ,m-2.$$

(53)

These equations, along with conditions (50) and (51), determine the optimal solution.

When there is only one intermediary, and it resells the resource between two markets while contacting them continuously, i.e., $m=2$, $p_1=p_{+}$, $p_2=p_{-}$, the optimal prices of buying and selling are

$$U_1^{\ast }=\frac{2\cdot {\alpha }_{-}\cdot p_{-}+{\alpha }_{+}\cdot (p_{+}+p_{-})}{2\cdot ({\alpha }_{+}+{\alpha }_{-})},$$

(54)

$$U_2^{\ast }=\frac{2\cdot {\alpha }_{+}\cdot p_{+}+{\alpha }_{-}\cdot (p_{+}+p_{-})}{2\cdot ({\alpha }_{+}+{\alpha }_{-})},$$

(55)

$${\eta }_{\max }=\frac{U_1^{\ast }-U_2^{\ast }}{U_1^{\ast }}.$$

(56)

Example 2.

What is the maximum possible profit of an intermediary when trading between two markets, the first of which evaluates one unit of resource as USD 12, while the second one’s evaluation is USD 15?

The intensities of buying and selling are $g_{-}=0.7(U_1-12)$, $g_{+}=0.3(15-U_2)$.

Optimal prices of buying and selling, calculated using (22), are $U_1^{\ast }=12.45$, $U_2^{\ast }=13.95$. Its maximum profit is USD 1.5 per unit, and the profit-to-expenses ratio is ${\mathsf{\eta }}_{\max }=12\%$. The maximum profit stream is USD 0.47 per time unit.

For kinetic factors going to infinity, ${\mathsf{\eta }}_{\max }=p_{+}/p_{-}-1=25\%$.

For a microeconomic system with linear dependence of flows on the difference in evaluations, an analogue of Prigozhin’s extreme principle in thermodynamics is true, which is as follows: In an open system of economic agents, the stationary resources are allocated in a way that ensures that the capital dissipation reaches its minimum:

$$\sigma =\frac{1}{2}{\displaystyle \sum _{i,j=1}^m{\Delta }_{ij}^T\cdot A_{ij}\cdot {\Delta }_{ij}}+{\displaystyle \sum _{i=1}^m{\Delta }_i^T\cdot A_i\cdot {\Delta }_i}\to \underset{p}{\min },$$

(57)

$${\Delta }_{ij}=p_i-p_j, i,j=1,\dots ,m,$$

(58)

$${\Delta }_i=U_i-p_i, i=1,\dots ,m.$$

(59)

5. Optimal Processes in Macrosystems

The common problems of optimal control in MS are the problems of transitioning the system from one state to another with minimum cost or maximum extraction of “organized” resource. In thermodynamics, it is mechanical or electrical energy, the work of separation; in microeconomics, it is capital. The following are the results of solving some problems.

5.1. Thermodynamics

Work capacity is the maximum work that can be extracted from a non-equilibrium system in a given time $\tau$. Each subsystem is characterized by an initial state. Some constraints are imposed on the set of finite states. The heat converter can interact with each subsystem, changing its own intensive variables $U(t)$ so as to extract the maximum work. The controls are, in addition to these variables, contact functions equal to one during interaction and zero with no interaction, as well as the final state of the system taking into account the constraints.

The following statements are true [11]:

• In the optimal process, for each interaction of a converter with the finite-capacity subsystems, its intensive variables $U^{\ast }(t)$ must satisfy the minimum dissipation conditions.
• In a system consisting of reservoirs, for any heat and mass transfer laws, the vector function composed of $U^{\ast }(t)$ and contact functions is piecewise constant. The number of its values does not exceed $k+1$, where $k$ is the number of conditions imposed on the system state at $\tau$. The system entropy grows as a piecewise linear function.

Corollary: At $k=0$, the entropy in the optimal process grows at a constant rate, and the converter interacts with only one reservoir.

Results: For a system of n thermal subsystems of finite capacity $C_i$$(i=1,\dots ,n)$ and a reservoir with temperature $T_{-}$:

$$q_i={\alpha }_i\cdot (U_i-T_i), i=1,\dots ,n,$$

(60)

$$\frac{d{T}_i}{dt}=\frac{q_i}{C_i}, i=1,\dots ,n,$$

(61)

$$T_i(0)=T_{0i},$$

(62)

$$A_{\infty }={\displaystyle \sum _{i=1}^n{C}_i\cdot \left[T_{0i}-T_{-}\cdot \left(1+\ln \frac{T_{0i}}{T_{-}}\right)\right]},$$

(63)

$$A_{\tau }^{\ast }=Q_{+}(k)-Q_{+}(k),$$

(64)

$$Q_{+}(k)={\displaystyle \sum _{i=1}^n{T}_{0i}\cdot C_i\left[1-\exp \left(\frac{\tau \cdot {\alpha }_i\cdot (1-k_i)}{C_i}\right)\right]},$$

(65)

$$Q_{-}(k)={\displaystyle \sum _{i=1}^n\frac{\tau \cdot T_{-}\cdot {\alpha }_i\cdot {\alpha }_{-}\cdot (1-k_i)}{{\alpha }_{-}\cdot k_i-{\alpha }_i\cdot (1-k_i)}},$$

(66)

$$T_{0i}\cdot \exp \left(-\frac{\tau \cdot {\alpha }_i\cdot (1-k_i)}{C_i}\right)=\frac{T_{-}\cdot {\alpha }_{-}^2}{{\left({\alpha }_{-}\cdot k_i-{\alpha }_i\cdot (1-k_i)\right)}^2}, i=1,\dots ,n.$$

(67)

Similar solutions are true for a system without a reservoir.

Separation systems. Notation: $N_0$ is the number of moles of mixture, ${\gamma }_j$ is the mixture fraction in j-th vessel after separation, $x_{ij}$ is the concentration of the i-th component in j-th vessel, and ${\alpha }_{ij}=\frac{{\alpha }_{ij}\cdot {\alpha }_{i0}}{{\alpha }_{i0}+{\alpha }_{ij}}$ is the reduced mass transfer ratio calculated through the mass transfer ratio upon interaction of the converter and the i-th flow in the initial mixture ${\alpha }_{i0}$ and in the j-th flow at the outlet.

Minimum separation work with finite time $\tau$:

$$A_{\min }=R\cdot T\cdot N_0{\displaystyle \sum _{j=0}^m{\gamma }_j}{\displaystyle \sum _i^{}{x}_{ij}\ln \frac{x_{ij}}{x_{i0}}}+\frac{N_0^2}{\tau }{\displaystyle \sum _{j=1}^m{\gamma }_j^2}{\displaystyle \sum _j^{}\frac{x_{ij}^2}{{\alpha }_{ij}}}.$$

(68)

Figure 2 shows the first and second items of this sum, $A^0$ and $\Delta A$, respectively. The first summand is the reversible work of separation, and the second summand is the minimum cost due to irreversibility, $A_{\min }$, for a mixture of two components and complete separation. At zero and one, the second summand is missing. This explains why the reversible evaluation of the separation work (Gibbs’ separation work) gives a very large error for “poor” mixtures, such as in the separation of uranium isotopes.

5.2. Microeconomics

Profitability is the maximum capital that can be extracted in an EA system with different initial states over time $\tau$.

Each of m EAs obtains the initial stocks of resource and capital, and their corresponding evaluation $p_i(N_{i0},N_i)$. The system may or may not include a $p_{-}$ rated reservoir. The intermediary changes the prices of purchases (sales) $C_i(t)$, $(i=1,\dots ,m)$ so as to achieve the maximum profit.

$$\frac{d{N}_i}{dt}=-n_i(p_i,C_i), i=1,\dots ,m,$$

(69)

$$N_i(0)=N_i^0, i=1,\dots ,m,$$

(70)

$$\frac{d{N}_{0i}}{dt}=C_i\cdot n_i(p_i,C_i), i=1,\dots ,m,$$

(71)

$$N_{0i}(0)=N_{0i}^0, i=1,\dots ,m.$$

(72)

Profitability at $\tau \to \infty$ is similar to exergy in thermodynamics. It is equal to

$$E_{\infty }={\displaystyle \sum _{i=1}^n{\displaystyle \underset{N_i^0}{\overset{{\overline{N}}_i}{\int }}{p}_i(N_i,N_{0i})d{N}_i}},$$

(73)

where ${\overline{N}}_i$ in the system with the reservoir is determined by the equilibrium conditions (as the equation of prices in equilibrium to the reservoir prices)

$$p_i({\overline{N}}_i,{\overline{N}}_{0i})=p_{-}, i=1,\dots ,m.$$

(74)

If the system has no reservoir, the right part of equality (74) contains the equilibrium price $\overline{p}$. It is selected in such a way that the prices that depend on it would satisfy the condition of resource non-accumulation by the intermediary.

$${\displaystyle \sum _{i=1}^N\left[{\overline{N}}_i({\overline{p}}_i)-N_i^0\right]=0}.$$

(75)

At finite $\tau$, the same statement as in the maximum work problem is true, namely, prices $C(t)$ must vary so that minimum capital dissipation conditions are met when contacting any of the EAs (30). These conditions define the optimal price $C_i^{\ast }$ as a function of the current resource stock $N_i$ at the i-th EA and its final stock $N_i(\tau )={\overline{N}}_i\cdot C_i^{\ast }(N_i,{\overline{N}}_i)$.

Notably, the optimal price satisfies the condition

$$C_i^{\ast }(N_i,{\overline{N}}_i)+{\displaystyle \underset{N_i^0}{\overset{{\overline{N}}_i}{\int }}\frac{\mathsf{\partial }{C}_i^{\ast }}{\mathsf{\partial }{\overline{N}}_i}d{N}_i}=\Lambda , i=1,\dots ,m.$$

(76)

Also, the following equation is true:

$${\displaystyle \sum _{i=1}^n{\overline{N}}_i}={\displaystyle \sum _{i=1}^n{N}_i^0}.$$

(77)

6. Bounds of Feasible States of Macrosystems

Besides the direct restrictions on the state of MS imposed in a particular problem, these systems are restricted by the fact that in closed systems, the irreversibility index can only increase, while in open systems, the dissipation (energy, capital) is non-negative.

The general methodology for constructing the feasibility bounds for closed MS including active subsystems is as follows:

• Write down the balance equations, including the balance relation of the irreversibility factor $S$. For thermodynamic systems, these are the equations of material energy and entropic balance. For microeconomic systems, there are balances of resources, capital, and welfare function. The equation for the irreversibility factor involves dissipation $\sigma \ge 0$. Solving this equation with respect to $\sigma$, and taking into account the non-negative value of this variable, brings an inequality that limits the feasibility bounds. One bound $D_0$ at $\sigma$ = 0 corresponds to reversible processes.
• Under the restrictions imposed on the duration of the process, find a minimum value $\sigma ={\sigma }_{\min }$ at which one state or another can be achieved. This value corresponds to the minimum dissipation process. Instead of the inequality $\sigma \ge 0$, the balance relation of $S$ includes the condition $\sigma \ge {\sigma }_{\min }(\tau )$. One feasibility bound is dissipation, which is not zero but its minimum value. It narrows the bounds of feasible states to $D(\tau )\subset D_0$.

This can be demonstrated by the feasibility bounds of the heat machine in coordinates: heat flux from a working body $q_{+}$—power $p$ (Figure 3).

The reversible feasibility bounds ($\tau \to \infty$, $p\to 0$) are not actually “bound” and lie under the straight line, the slope of which is equal to the Carnot efficiency.

$${\eta }_k=1-\frac{T_{-}}{T_{+}}.$$

(78)

The efficiency for an irreversible heat machine with power $p$ does not exceed its value at its maximum $D$ (see [9]).

$$\eta (p)\le \frac{1}{2}\left({\eta }_k+\frac{p}{\overline{\alpha }\cdot T_{+}}\right)+\sqrt{\frac{1}{4}{\left({\eta }_k+\frac{p}{\overline{\alpha }\cdot T_{+}}\right)}^2-\frac{p}{\overline{\alpha }\cdot T_{+}}}.$$

(79)

Similarly, feasibility bounds can be made for separation processes, heat exchange systems, etc.

The feasibility set of microeconomic systems is determined by the balance relationships of commodity resources and capital, and the requirement that capital dissipation $\sigma$ is not less than the corresponding optimal process under given constraints. This inequality at $\sigma =0$ corresponds to processes of arbitrarily long duration or to demand and supply functions with arbitrarily large slopes. In other cases, ${\sigma }_{\min }$ depends on $\tau$ or the intensity of the processes.

For example, for an intermediary between two markets with resource valuations $p_{+}$ and $p_{-}$, with the feasibility bounds in the coordinates of the purchase costs, the profit flow looks similar to Figure 3.

The slope $D_0$ equals ${\eta }_0=\frac{p_{+}}{p_{-}}-1$. The marginal profit intensity (similar to $p_{\max }$) is

$$m_{\max }=\frac{\sqrt{{\alpha }_1\cdot {\alpha }_2}}{4\cdot \left(\sqrt{{\alpha }_1}+\sqrt{{\alpha }_2}\right)}\left[\sqrt{{\alpha }_2}\cdot (p_{+}^2-{\lambda }^2)+\sqrt{{\alpha }_1}\cdot (p_{-}^2-{\lambda }^2)\right],$$

(80)

$$\lambda =\frac{p_{-}\cdot \sqrt{{\alpha }_1}+p_{+}\cdot \sqrt{{\alpha }_2}}{\sqrt{{\alpha }_1}+\sqrt{{\alpha }_2}},$$

(81)

where ${\alpha }_1$ and ${\alpha }_2$ are the slopes of the linear functions of supply and demand.

Example 3.

Let us find the maximum profit stream of an intermediary for data from Example 2 while buying and selling the resource sequentially. In that case, $p_{+}=15$, $p_{-}=12$, ${\alpha }_1=0.7$, and ${\alpha }_2=0.3$. Equation (80) leads to $m_{\max }=0.12$ USD as the maximum profit stream of the intermediary, which is significantly lower than while continuously interacting with the markets of sellers and buyers.

7. Conclusions

This study considered the problems of optimizing the organization of contacts between subsystems in thermodynamics and economics, in particular, the problems of the maximum intensity of extracting a basic resource (work or money) from the system. In thermal machines, the entropies at the input and output of a working body are the same, while the heat streams (products of entropy streams and absolute temperature) are different. Similarly, in economics, the resource streams bought and sold by an intermediary during resource exchange are the same, while the streams of capital used to buy and earned when sold (products of resource streams and prices) are different. At the same time, the processes with non-zero intensities of streams cannot be reversed without carrying out work or spending capital.

To complete the analogy, it required a function that would serve as a quantitative valuation for the irreversibility factor. The existence of this function itself and its properties leads to important conclusions. One of them, in particular, is the dependence of the price in equilibrium of an i-th resource on the change in the reserves of a j-th resource, as well as the price of a j-th on the change in an i-th.

One of Carnot’s achievements was showing that energy as heat is different from energy as mechanical work. Transforming heat into work leads to diverting some of it into a cold source.

In a similar vein, capital in economics exists in a pure form (money, securities) and in a form of investment into material resources. When transforming invested capital into pure capital with non-zero streams, losses are inevitable, and they are similar to the losses that occur when transforming heat into work. An important difference between the exchange processes in economics and similar ones in thermodynamics is the “voluntary principle”. This means that the wealth function grows not just for the whole system, but for each of its subsystems that participates in the exchange.

The above results demonstrate the similarity of problems in thermodynamic and microeconomic macrosystems in the class of irreversible processes. These results show that it is possible to use the methodology of thermodynamic optimization in solving economic problems, particularly substance, energy, and entropy balance problems, as well as minimum irreversibility factor growth (dissipation) at limited stream intensities and kinetic factors.

This paper also shows that, economically speaking, the increase in the welfare function is the increase in trade costs. This study derived the conditions of minimum dissipation in economics and the equations for optimal buying/selling prices. Additionally, it determined the equations for the distribution of resources within stationary open microeconomic systems and feasibility bounds in a system with an intermediary.

An orchestration of processes when the dissipation is at the lowest possible is optimal for most natural criteria both in thermodynamics and microeconomics.